The strong employment report for June is likely to add fuel to the debate within the Federal Reserve about when to start raising short-term interest rates from near zero, and prompted several analysts to say the central bank could move sooner than they had previously anticipated.

J.P. Morgan Chase Bank economist Michael Feroli said in a note that building job-market momentum along with signs of improvement in other parts of the economy mean the central bank will likely start rate increases in the third quarter of next year, rather than in the final three months of 2015.

“We are pulling forward our projection for Fed tightening (the first time we have done so in recent memory),” he wrote.

Capital Economics’ chief U.S. economist, Paul Ashworth, wrote that the strong job growth figures are among “the key reasons why we think the Fed will be persuaded to begin raising interest rates earlier than most expect.” He forecast the first rate hike in March.

Bank of America’s co-head of global economics research, Ethan Harris, wrote, “This report suggests an increased risk of an earlier first rate hike, but Fed officials will likely take time to reassess.”

No Fed officials spoke Thursday after the new data were released, but most of them—13 of 16 individuals—indicated in June they expected to start raising their benchmark short-term rate next year. One Fed official expected the first move to come this year, while two forecast it for 2016, reflecting the breadth of views on the matter among the decision makers. The individual forecasts are anonymous.

In a speech last week in Puerto Rico, New York Fed President William Dudley allowed that while there’s a lot of uncertainty in the outlook, the market view that the central bank won’t increase rates until the middle of next year “sounds to me like a reasonable forecast” given the expected path of the economy.

However, St. Louis Fed President James Bullard warned recently that if the economy continues to pick up speed, it’s possible the Fed will need to lift rates earlier in 2015 than many officials now predict.

Mr. Bullard has said he believes the Fed will start raising rates near the end of the first quarter of next year. He said in New York late last month, “I don’t think financial markets have internalized how close we are to our ultimate goals, and I don’t think the [Fed’s policy-making committee] has internalized how close we are to our ultimate goals.”

If the economy continues to improve, “I predict the conversation about monetary policy will change,” Mr. Bullard said

The Labor Department reported Friday employers added 288,000 jobs in June, the fifth consecutive month of gains over 200,000—a streak unmatched since the end of the 1990s.

Unemployment fell to 6.1% from 6.3% the month prior, faster than Fed officials had expected. Just last month, they projected the jobless rate to come in between 6% and 6.1% in the fourth quarter this year. In the wake of the jobs data, a number of economists said it won’t take much for the jobless rate to fall further.

Mr. Feroli said “the inexorable decline in the unemployment rate, alongside firming core [personal consumption expenditures price index] inflation, is dramatically reducing the degree to which the Fed is missing on its mandate.” In that environment, it’s more likely that the Fed will have to start the process of normalizing the stance of short-term interest rates sooner, he said. Mr. Feroli expects to see the benchmark federal funds rate at 1% by the end of next year, and at 2.5% by the close of 2016, up from near zero now.

Fed officials’ views on interest rates are likely to shift only if more data come in showing the economy gaining greater strength than they anticipated. The economy shrank at an annual rate of 2.9% in the first quarter and appears to have rebounded somewhat in the second quarter. Many policymakers will likely want to see more confirmation that the first quarter was an aberration before changing their interest-rate plans.

“While today’s report pushes forward … timing of the first Fed rate hike, it is unlikely to happen before early 2015,” said Cary Leahey, an economist with Decision Economics. “It is hard to imagine the Fed hiking rates within six to nine months of the sizeable drop” in first-quarter gross domestic product.

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